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Uniform persistence and flows near a closed positively invariant set. (English) Zbl 0811.34033
The behavior of a continuous flow in the vicinity of a closed positively invariant set in a metric space is studied. The obtained results generalize results of Ura-Kimura and Bhatia on classification of a flow near a compact invariant set in a locally compact metric space. Applying the obtained results, the authors prove two persistence theorems. One of the theorems unifies and generalizes earlier persistence results based on Lyapunov-like functions and those about the equivalence of weak uniform persistence and uniform persistence. The second theorem generalizes persistence results based on analysis of a flow on the boundary by relaxing point dissipativity and invariance of the boundary. The obtained results are illustrated by considering several ecological systems.
MSC:
37-99Dynamic systems and ergodic theory (MSC2000)
37C10Vector fields, flows, ordinary differential equations
34D05Asymptotic stability of ODE
34G20Nonlinear ODE in abstract spaces
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